Money, Time and Interest Rates

Principles of present and future value, discounting, annuities, and valuation applications

30 min read
Beginner

Introduction

From this lesson onward, we will view money as a dynamic resource – one whose value and role evolve over time.

You will see how the same amount of money can represent either an opportunity (when investing) or a cost (when borrowing), depending on timing and context. Rather than treating interest as a formula to memorize, we will approach it as a price for time and risk.

You will learn how different compounding structures affect outcomes, why investment horizons matter, and how interest rates reflect human preferences and economic uncertainty. The mathematics involved is straightforward and serves only to support the underlying intuition.

Money and Time

One of the foundational ideas in finance is that money today is not equivalent to money in the future. This concept, known as the time value of money, underpins nearly every financial decision, from saving and investing to borrowing and pricing assets.

At its core, the idea rests on two simple observations:

  • Money today can be put to work.

If you have $1,000 now, you can invest it and earn a return. Even at a modest rate, that $1,000 grows over time. Because future money can be generated from present money, receiving funds sooner provides more flexibility and opportunity.

  • People value immediacy

Individuals generally prefer consumption now rather than later. Choosing to delay spending requires compensation. That compensation comes in the form of additional future purchasing power, which is precisely what interest provides.

From a financial perspective, delaying consumption only makes sense if the future reward is sufficiently larger than the present sacrifice. This trade-off between now and later is the essence of financial decision-making.

Interest as the Price of Time and Risk

Interest rates exist to balance the incentives of borrowers and lenders.

  • For borrowers, interest represents the cost of accessing money earlier than they otherwise could.
  • For savers and investors, interest is the return required to justify postponing consumption.

Every loan involves two components:

  • The principal, which is the original amount borrowed.
  • The interest, which compensates the lender over time.

Interest rates are usually quoted annually, but their true meaning depends on duration, compounding structure, and risk.

Why Interest Rates Differ

Interest rates are not arbitrary. They reflect several underlying factors:

  • Time horizon: Longer-term loans typically carry higher rates because uncertainty increases as time extends.

  • Risk of non-repayment: Lenders demand higher returns when the probability of default is greater. A new business venture, for example, is riskier than a stable salary-backed loan.

  • Economic conditions: Inflation expectations, recessions, and overall financial stability influence how much compensation lenders require.

  • Market competition: Financial institutions compete for borrowers and investors, which can push rates up or down depending on supply and demand for capital.

Seen this way, interest rates are not just numbers – they are signals that summarize time preferences, risk assessments, and broader economic forces.

Valuing Money Across Time: Present and Future Values

To make sound financial decisions, we must be able to compare cash flows that occur at different points in time. A dollar received today, a dollar received next year, and a dollar received ten years from now are not directly comparable unless we translate them into a common time frame.

This translation process is the mathematical core of finance. It relies on how interest is calculated and accumulated over time.

At a high level, there are two ways interest can be applied:

  • Simple interest

  • Compound interest

Understanding the difference between them is essential, because even small differences in structure can lead to large differences in outcomes over longer horizons.

Simple Interest: Linear Growth

Under simple interest, interest is calculated only on the original amount invested or borrowed (the principal). The interest earned each year is constant and does not build on itself.

As a result, interest grows linearly with time: previously earned interest does not itself generate additional interest.

If:

  • PP is the principal,
  • rr is the annual interest rate,
  • tt is the number of years,

then the future value under simple interest increases in direct proportion to time.

simple interest=P×r×t\text{simple interest} = P \times r \times t

While simple interest is easy to understand and compute, it is rarely used in long-term financial arrangements because it does not reflect how capital typically accumulates in real-world financial settings, where compounding effects are the norm.

Then, future value can be calculated as:

Future Value=P+simple interest=P+(P×r×t)\text{Future Value} = P + \text{simple interest} = P + (P \times r \times t)

Example: Simple Interest Calculation

Suppose you invest $1,000 at a simple interest rate of 5% per year for 3 years.

  • Principal (PP) = 1,000
  • Annual interest rate (rr) = 5% or 0.05
  • Time (tt) = 3 years

The total interest earned is:

simple interest=P×r×t=1,000×0.05×3=150\begin{aligned} \text{simple interest} &= P \times r \times t \\\\ &= 1{,}000 \times 0.05 \times 3 \\\\ &= 150 \end{aligned}
Simple Interest Calculation in Python
python
# Calculate simple interest
P = 1000
r = 0.05
t = 3
simple_interest = P * r * t
print(simple_interest)  # 150.0

Each year, the investment earns the same $50 in interest regardless of how much interest has already been earned.

After 3 years, the future value of the investment is:

Future Value=1000+150=1,150\text{Future Value} = 1000 + 150 = 1,150

This example illustrates the linear nature of simple interest: the interest earned increases by a fixed amount each year, and previously earned interest does not contribute to future interest.

Compound Interest: Exponential Growth

Under compound interest, interest is calculated on both the original principal and all interest previously earned. In effect, interest generates additional interest over time.

This produces a fundamentally different pattern of growth:

  • Returns compound, causing growth to accelerate rather than increase by a fixed amount each period.
  • Time becomes a decisive factor, as the compounding effect strengthens the longer capital remains invested.

With the same variables PP, rr, and tt, compound interest results in exponential growth rather than linear growth.

compound interest=P(1+r)t−P\text{compound interest} = P (1 + r)^{t} - P

This mechanism underlies the power of long-term investing – and, conversely, explains why long-term borrowing can become disproportionately expensive.

For calculating future values, the formula becomes:

Future Value=P(1+r)t\text{Future Value} = P (1 + r)^{t}

Example: Compound Interest Calculation

Suppose you invest $1,000 at an annual compound interest rate of 5% for 3 years.

  • Principal (PP) = 1,000
  • Annual interest rate (rr) = 5% or 0.05
  • Time (tt) = 3 years

The future value of the investment is:

Future Value=P(1+r)t=1,000×(1+0.05)3=1,000×1.157625=1,157.63\begin{aligned} \text{Future Value} &= P(1 + r)^t \\\\ &= 1{,}000 \times (1 + 0.05)^3 \\\\ &= 1{,}000 \times 1.157625 \\\\ &= 1{,}157.63 \end{aligned}

The total compound interest earned is:

compound interest=Future Value−P=1,157.63−1,000=157.63\begin{aligned} \text{compound interest} &= \text{Future Value} - P \\\\ &= 1{,}157.63 - 1{,}000 \\\\ &= 157.63 \end{aligned}
Compound Interest Calculation in Python
python
# Calculate compound interest (annual compounding)
P = 1000
r = 0.05
t = 3

FV = P * (1 + r) ** t   # Future Value
compound_interest = FV - P
print(FV)                 # 1157.625
print(compound_interest)  # 157.625

Why Compounding Dominates Over Time

Suppose you deposit $10,000 for 10 years in two different banks.

  • Bank A advertises a higher annual rate of 6%, but interest is calculated using simple interest.
Future Value=P+P×r×t=10,000+10,000×0.06×10=10,000+6,000=16,000\begin{aligned} \text{Future Value} &= P + P \times r \times t \\\\ &= 10{,}000 + 10{,}000 \times 0.06 \times 10 \\\\ &= 10{,}000 + 6{,}000 \\\\ &= 16{,}000 \end{aligned}
  • Bank B applies a 5.75% annual rate with compounding, allowing interest to earn interest over time.
Future Value=P(1+r)t=10,000×(1+0.0575)10≈10,000×1.744≈17,440\begin{aligned} \text{Future Value} &= P (1 + r)^t \\\\ &= 10{,}000 \times (1 + 0.0575)^{10} \\\\ &\approx 10{,}000 \times 1.744 \\\\ &\approx 17{,}440 \end{aligned}

Even though Bank A advertises the higher rate, Bank B ultimately delivers the larger final balance. The reason is not the rate itself, but the structure of accumulation. Compounding increasingly amplifies differences as time passes.

  • From the depositor’s perspective, compounding is beneficial.
  • From the borrower’s perspective, compounding increases the total cost of borrowing.

Time, not just rate, determines the outcome.

If the deposit period were only 1 year, however, there would be little or no difference between the two banks, because compounding requires time to have an effect.

Interest Rates as Returns and Costs

An interest rate plays a dual role in finance, depending on perspective.

From the borrower's point of view, it represents the cost of borrowing money: the price paid for having access to funds today rather than in the future. Higher interest rates make borrowing more expensive, while lower rates make it cheaper.

From the lender's or investor's point of view, the same interest rate represents the required return for giving up money today. It compensates the lender for delaying consumption, as well as for risks such as default and inflation.

For now, we assume interest rates are fixed over time. In reality, many financial contracts involve floating or reference-based rates (such as rates tied to LIBOR or other market rates), but these introduce additional complexity that we will address later.

Present Value and Future Value

We've calculated future value (FV) in the previous sections. The concept of present value (PV) answers a fundamental question: What is a future cash flow worth today?

Under compound interest, money has a time value: a dollar today is worth more than a dollar in the future because today’s dollar can be invested to earn interest.

As a result, future cash flows must be discounted back to the present to determine their current worth.

If:

  • FVFV is the future value,
  • rr is the annual interest rate,
  • tt is the number of years,

then the present value is given by:

Present Value=FV(1+r)t\text{Present Value} = \frac{FV}{(1 + r)^{t}}

Example

Suppose you are promised $1,157.63 to be received in 3 years, and the appropriate annual discount rate is 5%.

What is the value of this future cash flow today?

Present Value=FV(1+r)t=1,157.63(1+0.05)3=1,157.631.157625≈1,000\begin{aligned} \text{Present Value} &= \frac{FV}{(1 + r)^t} \\\\ &= \frac{1{,}157.63}{(1 + 0.05)^3} \\\\ &= \frac{1{,}157.63}{1.157625} \\\\ &\approx 1{,}000 \end{aligned}

This calculation reverses the compounding process: it determines the amount that, if invested today at rate rr, would grow into the specified future value.

Present Value Calculation in Python
python
# Calculate present value (annual discounting)
FV = 1157.63
r = 0.05
t = 3

PV = FV / (1 + r) ** t
print(PV)  # 1000

Solving for the Rate of Return

Given what we invest today and what we receive in the future, what rate of return did we earn?

The compound interest formula can also be rearranged to answer a different question:

If:

  • PVPV is the present value,
  • FVFV is the future value,
  • tt is the number of years,

then the annual rate of return is given by:

Rate of Return (r)=(FVPV)1t−1\text{Rate of Return}\ (r) = \left( \frac{FV}{PV} \right)^{\frac{1}{t}} - 1

Suppose you invest $1,000 today and receive $1,250 after 2 years.

What annual rate of return does this investment imply?

Rate of Return=(FVPV)1t−1=(1,2501,000)12−1=(1.25)0.5−1≈0.1180\begin{aligned} \text{Rate of Return} &= \left( \frac{FV}{PV} \right)^{\frac{1}{t}} - 1 \\\\ &= \left( \frac{1{,}250}{1{,}000} \right)^{\frac{1}{2}} - 1 \\\\ &= (1.25)^{0.5} - 1 \\\\ &\approx 0.1180 \end{aligned}

This corresponds to an annual rate of return of approximately 11.8%.

Rate of Return Calculation in Python
python
# Calculate annual rate of return
PV = 1000
FV = 1250
t = 2

r = (FV / PV) ** (1 / t) - 1
print(r)  # 0.1180

The Yield Curve: Interest Rates Across Time Horizons

Interest rates generally depend on how long money is committed. To understand how interest rates vary across maturities at a given point in time, we use a tool known as the yield curve.

The yield curve plots the relationship between:

  • Time to maturity (on the horizontal axis), and
  • Interest rates (on the vertical axis)

[INSERT FIGURE HERE]

In most circumstances, short-term interest rates are lower than long-term interest rates. The intuition is straightforward: lending money for longer periods exposes investors to greater uncertainty, and they demand additional compensation for bearing that risk.

However, this pattern is not guaranteed. At times, short-term interest rates can equal or even exceed long-term interest rates. When this occurs, it reflects shifts in economic expectations, monetary policy, or perceived risk over different time horizons.

For example, if investors expect economic growth to slow or anticipate future reductions in policy rates, demand for long-term bonds may increase, pushing long-term yields down. At the same time, central banks may maintain high short-term rates to control inflation, causing short-term yields to rise. The result is a flattened or inverted yield curve.

Interpreting Yield Curve Shapes

Here's a quick overview of common yield curve configurations and their interpretations:

  • Upward-sloping (normal) yield curve: Long-term rates are higher than short-term rates. This reflects expectations of economic growth, inflation, and compensation for long-term risk.

  • Flat yield curve: Short- and long-term rates are similar, often signaling uncertainty or a transition in economic conditions.

  • Inverted yield curve: Short-term rates exceed long-term rates. This configuration suggests that investors view near-term conditions as riskier than the distant future.

Historically, an inverted Treasury yield curve has often preceded economic recessions. The intuition is that policymakers may lower short-term rates to stimulate economic activity when a slowdown is expected, while long-term rates already reflect weaker growth expectations.

Thus, the yield curve is not merely a descriptive tool, it is also an important economic signal.

Spot Rates and Forward Rates

So far, we have discussed interest rates as if investment begins immediately. In practice, investors frequently need to compare investments that begin at different future dates. This requires distinguishing between spot rates and forward rates.

A spot interest rate is the rate available today for an investment that starts immediately and lasts until a specified maturity.

For example:

  • A 1-year spot rate applies to money invested today and returned in one year.

  • A 2-year spot rate applies to money invested today and returned in two years.

Spot rates for different maturities are precisely what we observe on the yield curve.

Now consider a different situation. Suppose you do not want to invest today. Instead, you plan to invest starting one year from now, for a period of two years.

The spot rate for a 2-year investment is no longer relevant, because that rate applies to an investment starting today. What you need is the interest rate implied for a future investment period. This leads to the concept of a forward rate.

A forward rate answers the question: What annual return is implied today for an investment that will begin in the future and last for a specified length of time?

Connecting Spot Rates and Forward Rates

Forward rates are not quoted independently. They are implied by existing spot rates through the logic of no-arbitrage: investors should not be able to earn a risk-free profit by choosing one sequence of investments over another.

Using compound interest:

  • Investing directly for multiple years at a long-term spot rate
  • Or investing sequentially using shorter-term spot and forward rates

must produce the same final value.

From this equivalence, we can derive forward rates mathematically from observed spot rates.

Let:

  • sts_t be the tt-year spot rate,
  • sss_s be the ss-year spot rate, with s>ts > t,
  • ft,sf_{t,s} be the forward rate applicable from year tt to year ss.

Then no-arbitrage implies:

(1+ss)s=(1+st)t(1+ft,s)s−t(1 + s_s)^s = (1 + s_t)^t (1 + f_{t,s})^{s-t}

Solving for the forward rate:

1+ft,s=((1+ss)s(1+st)t)1s−tft,s=((1+ss)s(1+st)t)1s−t−1\begin{aligned} 1 + f_{t,s} &= \left( \frac{(1 + s_s)^s}{(1 + s_t)^t} \right)^{\frac{1}{s-t}} \\\\ f_{t,s} &= \left( \frac{(1 + s_s)^s}{(1 + s_t)^t} \right)^{\frac{1}{s-t}} - 1 \end{aligned}

Example

Suppose the 1-year spot rate is 4% and the 2-year spot rate is 5%. What 1-year forward rate starting one year from now is implied?

f1,2=((1+0.05)2(1+0.04)1)12−1−1=1.10251.04−1≈0.0601\begin{aligned} f_{1,2} &= \left( \frac{(1 + 0.05)^2}{(1 + 0.04)^1} \right)^{\frac{1}{2-1}} - 1 \\\\ &= \frac{1.1025}{1.04} - 1 \\\\ &\approx 0.0601 \end{aligned}

The implied 1-year forward rate is approximately 6.01%.

Forward Rate Calculation in Python
python
# Calculate implied forward rate from spot rates
s_t = 0.04  # t-year spot rate
s_s = 0.05  # s-year spot rate
t = 1
s = 2

f_t_s = ((1 + s_s)**s / (1 + s_t)**t)**(1 / (s - t)) - 1
print(f_t_s)  # 0.0601

Interpreting an Implied Forward Rate

When we compute a forward rate between two future dates, we are not forecasting what interest rates will be. Instead, we are identifying what rates are embedded in current market prices.

Thus, forward rates reflect:

  • Market expectations
  • Risk premiums
  • Time-value considerations

They are central to pricing bonds, managing interest rate risk, and evaluating future investment opportunities.

The same logic applies to forward rates covering any future interval. By combining spot rates of different maturities, we can infer the entire term structure of implied future interest rates.

While the notation can become dense, the underlying idea remains simple: All interest rates across time must be internally consistent.

Variable Interest Rates and Uncertainty

Up to this point, we have largely assumed that interest rates are fixed over the life of an investment or loan. In reality, many financial contracts use variable (or floating) interest rates, meaning the rate paid or received changes over time in response to market conditions.

A common example is a variable-rate certificate of deposit (CD), sometimes referred to as a flex CD. These instruments allow investors to earn interest in a relatively safe account, but the interest rate is tied to a reference rate that can move up or down during the contract’s life.

How a Variable Rate Works

A variable-rate CD typically pays an interest rate defined as:

Reference rate ± spread\text{Reference rate ± spread}

The reference rate might be a policy rate or market benchmark, while the spread reflects the bank’s margin.

As the reference rate changes, the interest earned on the CD adjusts accordingly. This means the investor does not know in advance exactly how much interest will be earned over time.

Suppose an investor opens a 24-month variable-rate CD whose interest rate is defined as the federal funds rate minus a fixed spread. At the time the account is opened, the reference rate is relatively low, so the CD initially earns a modest return. Over time, the reference rate rises, increasing the interest earned. Later, rates decline again, and the return falls accordingly.

Because the interest rate changes over time, the future value of the investment cannot be calculated with certainty in advance. Unlike fixed-rate contracts, variable-rate instruments introduce uncertainty into even simple valuation problems.

Choosing Between Fixed and Variable Rates

The choice between fixed and variable rates is ultimately a strategic decision:

  • Variable rates are attractive when investors expect interest rates to rise.
  • Fixed rates are preferable when investors expect rates to fall or want certainty.

However, expectations alone are not sufficient. Several interrelated factors must be considered:

  • Liquidity needs: Can the funds be locked up for the full term, or might early withdrawal be required?

  • Time horizon: Is the investment duration aligned with future income and expense needs?

  • Risk tolerance: How comfortable is the investor with uncertainty in returns?

  • Flexibility trade-offs: Would it be better to split funds across multiple maturities rather than committing everything to a single decision?

These questions arise not only for investments, but also for borrowing decisions. For example, borrowers typically prefer variable rates when they expect rates to decline, and fixed rates when they want predictable payments.

The key takeaway is that even simple financial products embed meaningful choices, and those choices are made under uncertainty.

Finance is not merely about computing values – it is about making informed decisions when outcomes are uncertain. Variable interest rates highlight this reality and foreshadow many of the challenges encountered in more advanced financial engineering and risk management.

Nominal Interest Rates vs. Real Interest Rates

So far, we have discussed interest rates as if money’s purchasing power were constant. In practice, this is not the case. Inflation changes what money can buy over time, and ignoring inflation can lead to misleading conclusions.

To address this, finance distinguishes between nominal and real interest rates.

  • Nominal interest rate: The stated or observed rate of return on an investment.

  • Real interest rate: The rate of return after adjusting for inflation, reflecting changes in purchasing power.

An investor may earn a positive nominal return and still be worse off in real terms if prices rise sufficiently.

Inflation and Purchasing Power

Inflation refers to a general increase in prices across the economy. When inflation rises, each unit of currency buys fewer goods and services.

Suppose an investor earns interest over a short period and feels wealthier because the account balance increased. If prices rose during that same period, the investor may find that the interest earned does not buy as much as expected. In extreme cases, the investor’s purchasing power may have actually declined.

The Fisher Relationship

To connect nominal returns, inflation, and real returns, economists use the Fisher relationship.

A commonly used approximation is:

Real interest rate ≈ Nominal interest rate - Inflation rate\text{Real interest rate ≈ Nominal interest rate - Inflation rate}

This approximation works well when inflation and interest rates are relatively low.

For example:

  • If the nominal return is 3% and inflation is 2%, the real return is approximately 1%.
  • If nominal and inflation rates are equal, the real return is roughly zero.
  • If inflation exceeds the nominal return, the real return becomes negative.

A negative real interest rate means that delaying consumption reduces purchasing power rather than increasing it.

The approximation above ignores compounding. When we account for compounding explicitly, the exact Fisher equation relates nominal rates, real rates, and expected inflation multiplicatively rather than additively.

While the precise formula is more accurate, the intuition remains the same:

What ultimately matters is not how much money you earn, but what that money can buy.

Why Real Rates Matter

Real interest rates are critical for:

  • Comparing investment opportunities across time
  • Evaluating long-term savings decisions
  • Understanding economic incentives to save or spend

In practice, investors, borrowers, and policymakers must always look beyond nominal figures to assess true economic outcomes.

Rate of Saving vs. Rate of Borrowing

Interest rates are rarely symmetrical. The rate you earn on a deposit is almost always lower than the rate you pay as a borrower.

For example, imagine you deposit $10,000 in a savings account earning 3% per year. That same bank might charge 7.5% if you borrow $10,000 for an investment. Borrowing via a credit card could be even more expensive, sometimes exceeding 18%, depending on credit history.

The reason for this gap is simple: banks operate like any other business. They pay relatively little for funds (your deposits) and charge more for loans. Depositors provide the raw material – capital – that banks then deploy to earn higher returns. In addition, banks manage risk, operational costs, and profit margins, which are reflected in the interest spread.

Banks also differentiate by term length. Short-term deposits are generally used to fund long-term loans. Understanding the relationship between the borrowing/lending term and the interest rate is key to grasping how banks price money.

Term to Maturity

The term to maturity is the length of time money is invested or loaned before the principal is returned.

  • A 3-month certificate of deposit (CD) has a 3-month term.
  • A 6-month CD has a 6-month term.

Some accounts, like savings accounts, have no fixed term – you can withdraw funds at any time – but they typically offer very low rates. As of today, typical savings account rates are often fractions of a percent.

The term to maturity directly influences the interest rate. Longer terms usually require higher interest rates to compensate investors for delayed access and increased uncertainty.

Why Longer Terms Usually Earn More

Two intuitive reasons explain why longer-term investments often pay more:

  1. Time preference: Most people prefer to have money now rather than later. Delaying access for longer periods deserves a higher reward, especially as inflation and economic uncertainty accumulate over time.

  2. Uncertainty: A 1-month deposit is relatively safe – the investor can easily monitor repayments and adjust. A 1-year loan carries more risk: the borrower’s circumstances, the economy, and other factors may change. The longer the horizon, the higher the potential for unexpected events.

Default Risk

Default risk is another factor influencing interest rates. It represents the chance that the borrower cannot repay the loan.

  • Short-term loans are less risky because repayment is due quickly.
  • Long-term loans carry more uncertainty. Over months or years, borrowers may face income loss, economic downturns, or unexpected expenses.

Higher default risk leads lenders to charge higher interest, ensuring that they are compensated for the additional uncertainty.

Problem A: Retirement Goal in Today’s Dollars

Robert, 25, wants to retire at the end of 2064 with $1,000,000 in today's dollars. He plans to contribute annually from 2024 to 2063. He estimates a discount rate of 3% and expects his investments to grow at 5% per year.

Question: How much should he deposit each year?

Solution Overview:

Calculate the future value of $1,000,000 in 2064 using the discount rate.

Determine the annual contribution needed to reach that future value, accounting for 5% investment growth.

Result: Robert must deposit approximately $25,718 per year to reach his target.

Problem B: Retirement Spending Plan

Robert revises his goal: he wants to spend $150,000 per year for 20 years after retiring, from 2065 to 2084. How much should he contribute annually from 2024 to 2063 to fund this plan?

Solution:

This requires computing the present value of the desired retirement withdrawals.

Spreadsheet tools are ideal for handling this multi-step problem, though it can be solved mathematically with annuity formulas.

Key Insight: Spreadsheets are practical tools for complex financial planning. They allow iterative calculations, scenario testing, and easy updates as assumptions change.

Key Takeaways

  • Interest rates differ for saving and borrowing due to bank margins, risk, and term length.

  • Longer-term investments generally earn higher rates to compensate for delayed access and uncertainty.

  • Default risk increases with time and borrower uncertainty.

  • Time value of money principles – PV, FV, and discounting – allow us to plan for long-term goals such as retirement.

📈

Foundations of Financial Science

A comprehensive guide to thinking probabilistically in equity markets – covering probability, statistics, portfolio theory, time series, and derivatives.